Problem 5

Two circles

G_1

and

G_2

are contained inside the circle

G

, and are tangent to

G

at the distinct points

M

and

N

, respectively.

G_1

passes through the center of

G_2

. The line passing through the two points of intersection of

G_1

and

G_2

meets

G

A

and

B

. The lines

MA

and

MB

meet

G_1

C

and

D

, respectively. Prove that

CD

is tangent to

G_2

Step 3 of 6: Locate the chord on the centers line

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In plain words

The common chord’s projection is fixed by the radii.

O_2V=\frac{r_2^2}{2r_1}

Detailed analysis

Let $V=AB\cap O_1O_2$ . The standard intersecting-chords/homothety computation for two circles with $O_2\in G_1$ gives $O_2V=r_2^2/(2r_1)$ . This is obtained by taking an intersection point of $G_1,G_2$ and using the similar triangles formed by the midpoint of the corresponding chord.